Optimal. Leaf size=69 \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{2} a B x^2+\frac{1}{5} x^5 (A c+b C)+\frac{1}{4} b B x^4+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7 \]
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Rubi [A] time = 0.0355215, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {1657} \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{2} a B x^2+\frac{1}{5} x^5 (A c+b C)+\frac{1}{4} b B x^4+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7 \]
Antiderivative was successfully verified.
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Rule 1657
Rubi steps
\begin{align*} \int \left (A+B x+C x^2\right ) \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a A+a B x+(A b+a C) x^2+b B x^3+(A c+b C) x^4+B c x^5+c C x^6\right ) \, dx\\ &=a A x+\frac{1}{2} a B x^2+\frac{1}{3} (A b+a C) x^3+\frac{1}{4} b B x^4+\frac{1}{5} (A c+b C) x^5+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7\\ \end{align*}
Mathematica [A] time = 0.0127978, size = 69, normalized size = 1. \[ \frac{1}{3} x^3 (a C+A b)+a A x+\frac{1}{2} a B x^2+\frac{1}{5} x^5 (A c+b C)+\frac{1}{4} b B x^4+\frac{1}{6} B c x^6+\frac{1}{7} c C x^7 \]
Antiderivative was successfully verified.
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Maple [A] time = 0., size = 58, normalized size = 0.8 \begin{align*} aAx+{\frac{aB{x}^{2}}{2}}+{\frac{ \left ( Ab+aC \right ){x}^{3}}{3}}+{\frac{bB{x}^{4}}{4}}+{\frac{ \left ( Ac+bC \right ){x}^{5}}{5}}+{\frac{Bc{x}^{6}}{6}}+{\frac{cC{x}^{7}}{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.94561, size = 77, normalized size = 1.12 \begin{align*} \frac{1}{7} \, C c x^{7} + \frac{1}{6} \, B c x^{6} + \frac{1}{4} \, B b x^{4} + \frac{1}{5} \,{\left (C b + A c\right )} x^{5} + \frac{1}{2} \, B a x^{2} + \frac{1}{3} \,{\left (C a + A b\right )} x^{3} + A a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.12176, size = 161, normalized size = 2.33 \begin{align*} \frac{1}{7} x^{7} c C + \frac{1}{6} x^{6} c B + \frac{1}{5} x^{5} b C + \frac{1}{5} x^{5} c A + \frac{1}{4} x^{4} b B + \frac{1}{3} x^{3} a C + \frac{1}{3} x^{3} b A + \frac{1}{2} x^{2} a B + x a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.11015, size = 65, normalized size = 0.94 \begin{align*} A a x + \frac{B a x^{2}}{2} + \frac{B b x^{4}}{4} + \frac{B c x^{6}}{6} + \frac{C c x^{7}}{7} + x^{5} \left (\frac{A c}{5} + \frac{C b}{5}\right ) + x^{3} \left (\frac{A b}{3} + \frac{C a}{3}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09284, size = 82, normalized size = 1.19 \begin{align*} \frac{1}{7} \, C c x^{7} + \frac{1}{6} \, B c x^{6} + \frac{1}{5} \, C b x^{5} + \frac{1}{5} \, A c x^{5} + \frac{1}{4} \, B b x^{4} + \frac{1}{3} \, C a x^{3} + \frac{1}{3} \, A b x^{3} + \frac{1}{2} \, B a x^{2} + A a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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